Problem 38 - Precalculus Student Solutions Manual 5th

Question

Solve each inequality. Give the solution set using interval notation. $$|7-3 x| \leq 4$$

Step-by-Step Solution

Answer

The solution set is \[1, \frac{11}{3} \].

Step 1: Understanding the absolute value inequality

The inequality given is \( |7 - 3x| \leq 4 \). An absolute value inequality of the form \( |a| \leq b \) can be written as \(-b \leq a \leq b\).

Step 2: Remove the absolute value

Apply the rule from Step 1: \(-4 \leq 7 - 3x \leq 4\).

Step 3: Solve the compound inequality

Split the compound inequality into two parts:1) \(-4 \leq 7 - 3x\), and 2) \(7 - 3x \leq 4\). Solve each part separately.

Step 4: Solve \(-4 \leq 7 - 3x\)

Subtract 7 from both sides: \(-4 - 7 \leq -3x\) which simplifies to \(-11 \leq -3x\).Divide by -3 (and reverse the inequality sign): \( \frac{-11}{-3} \geq x \). Simplify: \(\frac{11}{3} \geq x \) or \( x \leq \frac{11}{3}\).

Step 5: Solve \(7 - 3x \leq 4\)

Subtract 7 from both sides: \(7 - 7 - 3x \leq 4 - 7\) which simplifies to \(-3x \leq -3\).Divide by -3 (and reverse the inequality sign): \( x \geq 1 \).

Step 6: Combine the solutions

From Steps 4 and 5, the combined solution is \(1 \leq x \leq \frac{11}{3}\).

Step 7: Write the solution in interval notation

The solution in interval notation is \[1, \frac{11}{3} \].

Key Concepts

Interval Notation

Understanding interval notation is crucial when expressing the solution set for inequalities. Interval notation provides a condensed way to describe all numbers lying between a set of endpoints.
In interval notation, we use square brackets \[ ... \] to denote that an endpoint is included in the set (called a closed interval), and parentheses \( ... \) to signify that an endpoint is not included (called an open interval).
For example:

• \[1, 3\] means all numbers between 1 and 3, inclusive.


• \(1, 3\) means all numbers between 1 and 3, but not including 1 and 3.


• \(1, 3\] means including 3 but not 1.


• \[1, 3\) means including 1 but not 3.

To express the solution of an inequality like \(1 \leq x \leq \frac{11}{3}\) in interval notation, we use \[1, \frac{11}{3}\]. This tells us that \(x\) can be any number from 1 up to \( \frac{11}{3}\), including both endpoints.

Compound Inequalities

Compound inequalities involve two separate inequalities joined by the words 'and' or 'or'.
When two inequalities are connected by 'and', we are finding the values that satisfy both inequalities at the same time. These are also known as 'conjunctions'.
Let's break down the compound inequality from our problem: \(-4 \leq 7 - 3x \leq 4\).
This can be split into two separate inequalities:

• \(-4 \leq 7 - 3x\)


• \(7 - 3x \leq 4\)

Solving these separately, we can then combine our solutions. The solution to a compound inequality joined with 'and' is the intersection of the solutions to the individual inequalities, leading us to our final solution.In our exercise, solving \(-4 \leq 7 - 3x\) and \(7 - 3x \leq 4\), we arrived at the final solution \(1 \leq x \leq \frac{11}{3}\).

Solving Inequalities

Solving inequalities is a bit like solving equations, but there are crucial differences to keep in mind. Inequalities are statements about the relative size or order of two values.
Follow these general steps when solving inequalities:

• Isolate the variable just as you would in an equation.


• If you multiply or divide both sides of an inequality by a negative number, reverse the inequality sign.


• Combine like terms and simplify.

Let's look at one part of our solution process: solving \(-4 \leq 7 - 3x\).

• Step 1: Subtract 7 from both sides: \(-4 - 7 \leq -3x\) simplifies to \(-11 \leq -3x\).


• Step 2: Divide by -3 (and reverse the sign because we are dividing by a negative number): \(\frac{-11}{-3} \geq x\) simplifies to \( \frac{11}{3} \geq x\).

Applying this method to each part of the compound inequality helps us find the intersection of the solutions.
Always remember to check your steps and verify that the solution works in the original inequality.